Generally, a device with multiple supply/ground pins intends for those pins to be externally connected to the appropriate supply/ground. It is particularly important to connect the ground pins together using wide traces or polygons/planes to avoid ground loops. The datasheet, section 1.1, for the 328p indicates three specific power connection types: VCC, GND, and AVCC. Thus, all of the VCC pins should be connected together, the GND pins should be connected together with care for a low impedance connection, and the AVCC pin should be connected (with optional filter described in the datasheet) to VCC.
While this device shows a direct connection between supply/ground pins, some devices require connections between those pins externally and may be damaged if some pins are left floating. It is best to assume that pins of the same supply must be connected together unless specifically noted in the device datasheet. Multiple external connections allows more current to safely flow into/out of the device.
Additionally, considering decoupling: multiple supply pins allows multiple bypass capacitors (usually 1000pF to 0.1uF, depending on the application) to be placed around the device. For a low speed device with short connections between supply pins and an uninterrupted ground plane around the device, a single 0.1uF capacitor may suffice. Higher speed devices may require one capacitor per supply pin or even multiple sizes of capacitor on each supply pin.